Structure Preserving Algorithms And Discrete Optimal Control For Birkhoffian Systems

This thesis, taking the discretization of variational principles as its main thread, con-structs structure preserving algorithms for Birkhoffian systems, discrete variational differ-ence schemes for generalized Birkhoffian systems, structure preserving algorithms for con-strained Birkhoffian systems and discrete optimal control theory for Birkhoffian systems, viathe discrete variational technique.First, the dissertation studies the way to construct structure preserving algorithms forBirkhoffian systems. By direct discretization of the Pfaff–Birkhoff variational principle, thecorresponding discrete Birkhoffian equations are obtained. Identified as numerical schemesfor the original continuous system, the resulting discrete Birkhoffian equations automatical-ly preserve the symplectic structure. Since the preservation of symplecticity and variationalcharacter, structure preserving algorithms formulated for Birkhoffian systems perform moreefficiently in simulating the motion of dynamical systems. The numerical results of thesimple pendulum, the Lotka–Volterra system and the linear damped oscillator indicate thatthe structure preserving algorithms constructed here, compared with traditional differencemethods, have obvious advantages in convergence, stability and preservation of conservedquantities. In addition, structure preserving algorithms obtained by discretizing the varia-tional principle are more precise than those with the same order accuracy but obtained byother existing approaches.Next, the dissertation discusses the way to get difference schemes with high efficien-cy for generalized Birkhoffian systems. The Pfaff–Birkhoff variational principle is modifiedto be the Pfaff–Birkhoff–D’Alembert principle which can directly induce the generalizedBirkhoffian equations. Then by direct discretization of the Pfaff–Birkhoff–D’Alembert prin-ciple, the corresponding discrete generalized Birkhoffian equations are obtained consequent-ly. When satisfying some non-degeneracy conditions, the resulting equations then determinea kind of difference schemes for original continuous systems, named as discrete variationaldifference schemes. Taking account of the approximative variational structure of continuoussystems, discrete variational difference schemes simulate the motion of dynamical system-s more accurately than traditional ones, even though they are no longer strict symplecticalgorithms. Numerical tests of illustrative examples indicate that the discrete variational d-ifference schemes formulated for generalized Birkhoffian systems can not only simulate themotion of the dynamical system efficiently, but also predict the energy evolution of the sys-tem precisely. Then, applying discrete variational difference schemes for generalized Birkhoffian sys-tems to optimal control problems yields the discrete optimal control theory for Birkhoffiansystems. Through direct discretization of the objective functional, the control system andfixed endpoints conditions, the optimal control problem for Birkhoffian systems is trans-formed into a nonlinear optimization problem with finite dimension, whose constrains areexactly the discrete variational difference schemes previously constructed for generalizedBirkhoffian systems. The resulting optimization problem corresponding to discrete varia-tional difference schemes, compared with those derived from traditional difference schemes,is more faithful and consequently gives more accurate discrete optimal control. In addition,the obtained optimal control force can satisfy the requirement of practical problems as longas the time step is sufficiently small.Finally, this thesis discusses the construction of structure preserving algorithms for con-strained Birkhoffian systems. Being different from traditional styles, the corresponding con-ditional extremum problem is first discretized to design symplectic methods for constrainedBirkhoffian systems. As the derivant of the discretized conditional extremum problem, theresulting discrete constrained Birkhoffian equations are naturally symplectic. Before the op-eration, the obtained algorithms require not only the specification of an initial configurationof the simulated system but also a second configuration. However, it is usually difficult oreven sometimes impossible to come up with an accurate second configuration that strictlyfulfills the constraints. Therefore, a natural, efficient and reasonable method of initializationof simulations is developed correspondingly.